Spectral Dimension of Trees with a Unique Infinite Spine

Abstract

Using generating functions techniques we develop a relation between the Hausdorff and spectral dimension of trees with a unique infinite spine. Furthermore, it is shown that if the outgrowths along the spine are independent and identically distributed, then both the Hausdorff and spectral dimension can easily be determined from the probability generating function of the random variable describing the size of the outgrowths at a given vertex, provided that the probability of the height of the outgrowths exceeding n falls off as the inverse of n. We apply this new method to both critical non-generic trees and the attachment and grafting model, which is a special case of the vertex splitting model, resulting in a simplified proof for the values of the Hausdorff and spectral dimension for the former and novel results for the latter.

Random Walk on Random Infinite Looptrees

“…In parallel to the above-mentioned progress, mathematical physicists have developed generating function methods to calculate the spectral dimension of random trees, see e.g. [10,11,16,17,24]. The benefits of those methods are their simplicity but the disadvantages are that they do not apply as generally and give somewhat weaker results regarding the existence of d s .…”

Multigraph models for causal quantum gravity and scale dependent spectral dimension

“…derived under certain assumptions for fixed graphs [6]. Models for which the spectral and Hausdorff dimensions are well understood analytically include the GRT, random combs [7], random brushes [8] and non-generic trees [9,10]. In the mathematical literature percolation clusters have been studied intensively, see for instance [11].…”